particle question

[michael11]

Has anyone got any idea how to solve this?

A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t +1)i + (4 − t^2 )j.

ii. Find the value of t∗ at which the particle has greatest distance from (0, 0, 0).
Hint: optimising squared distance may be the simpler method here.
iii. Show that at position r(t ), the particles velocity is not perpendicular to r(t).

[HallsofIvy]

Has anyone got any idea how to solve this?

A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t +1)i + (4 − t^2 )j.

ii. Find the value of t∗ at which the particle has greatest distance from (0, 0, 0).
Hint: optimising squared distance may be the simpler method here.
iii. Show that at position r(t ), the particles velocity is not perpendicular to r(t).
Yes! Follow the hint given! The distance from (0,0,0) to r(t) is sqrt(r(t).r(t))- but, as the hint says, it will be easier to maximize its square: r(t).r(t)= (4t+1)^2+ (4- t^2)^2. Do you know how to find the maximum value of that?

As far as iii is concerned, do you know how to find the velocity vector?